Hydrostatic lower bound on planetary central pressure ($P_c\ge G{M}^2/(8\pi R^4)$)

The hydrostatic pressure support equation in enclosed-mass coordinate is $dP/dm=-Gm/(4\pi r^4)$. Since $r(m)\le R$, integration from surface pressure zero to the centre givesIf mass density decreases outward, the mean density within $r$ is at least the global mean, so $r(m)\le R(m/M{)}^{1/3}$. This improves the bound towith equality for uniform mass density. Centrally concentrated planets need larger central pressure to support their inner mass.